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            <h1>FNet: フーリエ変換によるトークンの混合</h1>
<p>これは、論文「<a href="https://papers.labml.ai/paper/2105.03824">FNet: トークンをフーリエ変換と混合する」<a href="https://pytorch.org">をPyTorchで実装したものです</a></a>。</p>
<p><em>この論文では、<a href="../mha.html"><a href="https://en.wikipedia.org/wiki/Discrete_Fourier_transform">セルフアテンション層を2つのフーリエ変換に置き換えてトークンを混合します</a></a>。</em><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">7</span><span class="mord">×</span></span></span></span></span>これは自己注意よりも効率的です。<a href="https://paperswithcode.com/method/bert">BERT</a> on <a href="https://paperswithcode.com/dataset/glue">GLUE</a> ベンチマークでは、自己注意よりもこれを使用した場合の精度の低下は約 92%</p> です。
<h2>2 つのフーリエ変換によるトークンの混合</h2>
<p>フーリエ変換を非表示次元 (埋め込み次元) に沿って適用し、次にシーケンス次元に沿って適用します。</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord coloredeq eqa" style=""><span class="mord" style=""><span class="mord coloredeq eqb" style=""><span class="mord mathcal coloredeq eqi" style="">R</span></span><span class="mord coloredeq eqb" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqb" style=""><span class="mord coloredeq eqc" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqh" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">seq</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord coloredeq eqc" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqc" style=""><span class="mord coloredeq eqe" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqh" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">hidden</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqe" style="">(</span><span class="mord coloredeq eqe" style=""><span class="mord mathnormal coloredeq eql" style="">x</span></span><span class="mclose coloredeq eqe" style="">)</span></span><span class="mord coloredeq eqc" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span><span class="mord coloredeq eqb" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span></span></span></span></span></span></span></p>
<p>ここで、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span></span></span></span></span>は埋め込み入力で、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqi" style=""><span class="mord mathcal" style="">R</span></span></span></span></span></span>フーリエ変換を表し、複素数の実数成分を表します。</p>
<p>これをPyTorchに実装するのはとても簡単です。たった1行のコードです。この論文では、事前に計算されたDFT行列を使用し、行列の乗算を行ってフーリエ変換を行うことを提案しています</p>。
<p>以下は<a href="experiment.html">、<a href="https://paperswithcode.com/dataset/ag-news">FNetベースのモデルを使用してAG</a> Newsを分類するためのトレーニングコードです</a>。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">41</span><span></span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Optional</span>
<span class="lineno">42</span>
<span class="lineno">43</span><span class="kn">import</span> <span class="nn">torch</span>
<span class="lineno">44</span><span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-1'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-1'>#</a>
            </div>
            <h2>FNet-ミックストークン</h2>
<p>このモジュールは単純に実装します <span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord coloredeq eqa" style=""><span class="mord" style=""><span class="mord coloredeq eqb" style=""><span class="mord mathcal coloredeq eqi" style="">R</span></span><span class="mord coloredeq eqb" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqb" style=""><span class="mord coloredeq eqc" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqh" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">seq</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord coloredeq eqc" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqc" style=""><span class="mord coloredeq eqe" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqh" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">hidden</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqe" style="">(</span><span class="mord coloredeq eqe" style=""><span class="mord mathnormal coloredeq eql" style="">x</span></span><span class="mclose coloredeq eqe" style="">)</span></span><span class="mord coloredeq eqc" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span><span class="mord coloredeq eqb" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span></span></span></span></span></span></span></p>
<p>このモジュールの構造は、<a href="../mha.html">標準的なアテンションモジュールと同様の構造になっているため</a>、簡単に交換できます。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">47</span><span class="k">class</span> <span class="nc">FNetMix</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span></pre></div>
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        <div class='docs doc-strings'>
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                <a href='#section-2'>#</a>
            </div>
            <p><a href="../mha.html">通常のアテンションモジュールには</a>、、、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqf" style=""><span class="mord text" style=""><span class="mord" style="">query</span></span></span></span></span></span></span>、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqj" style=""><span class="mord text" style=""><span class="mord" style="">key</span></span></span></span></span></span></span>マスクにさまざまなトークンを埋め込むことができます。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqg" style=""><span class="mord text" style=""><span class="mord" style="">value</span></span></span></span></span></span></span></p>
<p>同じ関数シグネチャに従うので、直接置換できます。</p>
<p>FNetミキシングの場合<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqf" style=""><span class="mord text" style=""><span class="mord" style="">query</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqj" style=""><span class="mord text" style=""><span class="mord" style="">key</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqg" style=""><span class="mord text" style=""><span class="mord" style="">value</span></span></span></span></span></span></span></span>、マスキングはできません。<code  class="highlight"><span></span><span class="n">query</span></code>
(<code  class="highlight"><span></span><span class="n">key</span></code>
と<code  class="highlight"><span></span><span class="n">value</span></code>
) の形はです<code  class="highlight"><span></span><span class="p">[</span><span class="n">seq_len</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">,</span> <span class="n">d_model</span><span class="p">]</span></code>
。</p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">60</span>    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">query</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">value</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">mask</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span></pre></div>
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    <div class='section' id='section-3'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-3'>#</a>
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            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqf" style=""><span class="mord text" style=""><span class="mord" style="">query</span></span></span></span></span></span></span>、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqj" style=""><span class="mord text" style=""><span class="mord" style="">key</span></span></span></span></span></span></span>、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqg" style=""><span class="mord text" style=""><span class="mord" style="">value</span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span></span></span></span></span>そしてトークンのミキシングではすべてが等しくなければなりません</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">72</span>        <span class="k">assert</span> <span class="n">query</span> <span class="ow">is</span> <span class="n">key</span> <span class="ow">and</span> <span class="n">key</span> <span class="ow">is</span> <span class="n">value</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-4'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-4'>#</a>
            </div>
            <p>トークンのミキシングはマスキングをサポートしていません。つまり、すべてのトークンに他のすべてのトークンの埋め込みが表示されます。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">74</span>        <span class="k">assert</span> <span class="n">mask</span> <span class="ow">is</span> <span class="kc">None</span></pre></div>
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    <div class='section' id='section-5'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-5'>#</a>
            </div>
            <p><code  class="highlight"><span></span><span class="n">x</span></code>
わかりやすいように割り当てる</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">77</span>        <span class="n">x</span> <span class="o">=</span> <span class="n">query</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-6'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-6'>#</a>
            </div>
            <p>フーリエ変換を非表示 (埋め込み) 次元に沿って適用します <span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqe" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqh" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">hidden</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eql" style="">x</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
<p><a href="https://pytorch.org/docs/stable/complex_numbers.html">フーリエ変換の出力は複素数のテンソルです</a>。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">84</span>        <span class="n">fft_hidden</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span></pre></div>
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    <div class='section' id='section-7'>
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                <a href='#section-7'>#</a>
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            <p>シーケンスの次元に沿ってフーリエ変換を適用します <span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord coloredeq eqc" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqh" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">seq</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord" style=""><span class="mord coloredeq eqe" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqh" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">hidden</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqe" style="">(</span><span class="mord coloredeq eqe" style=""><span class="mord mathnormal coloredeq eql" style="">x</span></span><span class="mclose coloredeq eqe" style="">)</span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span></span></span></span></span></span></p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">87</span>        <span class="n">fft_seq</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">fft_hidden</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span></pre></div>
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    </div>
    <div class='section' id='section-8'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-8'>#</a>
            </div>
            <p>本物のコンポーネントを手に入れよう <span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord coloredeq eqb" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqi" style="">R</span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord" style=""><span class="mord coloredeq eqc" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqh" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">seq</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord coloredeq eqc" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqc" style=""><span class="mord coloredeq eqe" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqh" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">hidden</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqe" style="">(</span><span class="mord coloredeq eqe" style=""><span class="mord mathnormal coloredeq eql" style="">x</span></span><span class="mclose coloredeq eqe" style="">)</span></span><span class="mord coloredeq eqc" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">91</span>        <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">fft_seq</span><span class="p">)</span></pre></div>
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